Finite Group Databasebeta

← #14460047

Subgroups — #14460047

Order 1920.

772 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#5732C211yesmaximal, characteristic, complemented
2#11752532C211yesmaximal, characteristic
3#11751194C211yesmaximal, characteristic, complemented
4C3×#65376C211yesmaximal, characteristic, complemented
5#11747514C212yesmaximal, complemented
6#11747502C212yesmaximal, complemented
7#11747447C212yesmaximal, complemented
8#11747176C212yesmaximal, complemented
9#11747101C211yesmaximal, characteristic, complemented
10C4×#6655C211yesmaximal, characteristic, complemented
11#11746998C211yesmaximal, characteristic, complemented
12#10615395—31nomaximal, Hall
13C2×#6765C2212yes—
14C3×C5×C4⋊Q8C2211yescharacteristic
15#91758C2211yescharacteristic
16C5×#947—21no—
17C5×Q8×Dic3—21no—
18C5×#898—22no—
19C5×#897C2212yes—
20C5×#892C2212yes—
21C4×C5×D12C2211yescharacteristic
22C4×C5×Dic6C2211yescharacteristic
23C3×C4×D20C2211yescharacteristic
24#91449—21no—
25#91414—22no—
26D5×C4⋊Dic3—21no—
27#91365C2212yes—
28C2×#6663C2212yes—
29C2×#6655C2211yescharacteristic
30C2×D3×#168C2212yes—
31#91094C2212yes—
32#91094C2212yes—
33C4×C3⋊#168C2211yescharacteristic
34C3×#3941C2212yes—
35C3×#3939C2212yes—
36C3×C4⋊#168C2211yescharacteristic
37#90986C2211yescharacteristic
38#90975C2212yes—
39#90972C2211yescharacteristic
40#90964C2211yescharacteristic
41(C2×D3)⋊#168C2212yes—
42#80981—51nomaximal, Hall
43C5×#556—31no—
44#65885—31no—
45#65382—31no—
46#65376D311yescharacteristic, complemented
47#65340—32no—
48#65336—32no—
49#65320—31no—
50C4×#688—31no—
51D4×#168—31no—
52#65291—32no—
53#65285—32no—
54C2×C3×C5×Q8—21no—
55C2×Dic30C2312yes—
56C2×C5×D12D411yescharacteristic
57C2×C4×C5×D3C2312yes—
58C2×C5×Dic6C2311yescharacteristic
59C2×C3×D20C2311yescharacteristic
60C2×C3×C4×D5—21no—
61C2×#1201—22no—
62C2×D5×Dic3—22no—
63#6765—24no—
64#6765—42no—
65C3×C5×C4⋊C4—21no—
66C3×C5×C4⋊C4C2312yes—
67C3×C42×C5C2311yescharacteristic
68C4⋊Dic15—21no—
69C4⋊Dic15—21no—
70C2×C3⋊#168C2311yescharacteristic
71C2×C3⋊#168C2311yescharacteristic
72C5×(C2×D3)⋊C4—22no—
73C5×(C2×D3)⋊C4—22no—
74C5×(C2×D3)⋊C4D412yes—
75C5×C4⋊Dic3—22no—
76C5×C4⋊Dic3D411yescharacteristic
77C5×C4⋊Dic3C2311yescharacteristic
78C5×Dic3⋊C4—21no—
79C4×C5×Dic3—21no—
80C4×C5×Dic3—22no—
81C3×#685—21no—
82C3×C4⋊Dic5C2311yescharacteristic
83C2×C3×#168C2312yes—
84#6668—22no—
85(C2×D3)⋊Dic5—22no—
86#6663—42no—
87#6655—22no—
88D3×#168—42no—
89#5732D511yescharacteristic, complemented
90#5555—51no—
91C3×#474—51no—
92#5093—51no—
93C4×#214—51no—
94#4888—52no—
95#4887—52no—
96#4849—51no—
97#4834—51no—
98#4783—52no—
99#4772—52no—
100C2×#709—32no—
101C5×C4⋊Q8C2×D311yescharacteristic
102C5×#126—62no—
103C5×#124—61no—
104C5×C22⋊Q8—32no—
105C4×C5×Q8—31no—
106C4×C5×Q8—61no—
107C4×C5×D4—31no—
108C5×#117—32no—
109C2×#689—32no—
110C2×#688—31no—
111#4084—61no—
112#4070—32no—
113C22×#168—32no—
114D5×C4⋊C4—61no—
115#4029—62no—
116C4×D20C2×D311yescharacteristic
117#4015—31no—
118#3967—31no—
119C22⋊#168—32no—
120#3942—32no—
121#3942—32no—
122#3942—32no—
123#3941—31no—
124#3941C2×D312yes—
125#3940—31no—
126#3939C2×D312yes—
127C4⋊#168C2×D311yescharacteristic
128C4×#168—31no—
129#3302—151noSylow
130C22×C5×D3C2×D412yes—
131C22×C3×D5—21no—
132C3×C5×Q8—42no—
133C2×C3×C4×C5—22no—
134C2×C3×C4×C5C2×D411yescharacteristic
135C2×C3×C4×C5C2×D411yescharacteristic
136C2×C3×C4×C5C2411yescharacteristic
137C2×Dic15—22no—
138Dic30—22no—
139Dic30—22no—
140C2×C5×Dic3—21no—
141C2×C5×Dic3—21no—
142C2×C5×Dic3C2×D412yes—
143C5×D12—22no—
144C4×C5×D3—22no—
145C4×C5×D3—22no—
146C5×Dic6—21no—
147C5×Dic6—21no—
148C2×C3×Dic5—21no—
149C3×D20—21no—
150C3×D20#4112yes—
151C3×C4×D5—42no—
152#1201—44no—
153D5×Dic3—44no—
154C3⋊#168—21no—
155C3⋊#168—21no—
156C3×#168—22no—
157C2×#237—52no—
158C3×C4⋊Q8C2×D511yescharacteristic
159C3×C4×D4—51no—
160#947—21no—
161Q8×Dic3—21no—
162C2×#214—51no—
163#930—101no—
164C2×C4⋊Dic3—101no—
165C2×Dic12—52no—
166C2×D3×C8—52no—
167#898—22no—
168#897—52no—
169#897C2×D512yes—
170#892C2×D512yes—
171#886—102no—
172C4×D12C2×D511yescharacteristic
173#869—51no—
174C4×Dic6C2×D511yescharacteristic
175C3×#107—52no—
176C3×C4⋊C8—51no—
177C3×#102—52no—
178#832—51no—
179(C2×D3)⋊C8—52no—
180#816—52no—
181#810—51no—
182#810—52no—
183#810—52no—
184#808—51no—
185C4×C3⋊C8—51no—
186C2×C5×Q8—21no—
187C2×C5×Q8—31no—
188C2×C5×D4—31no—
189C22×C4×C5—32no—
190C2×C22⋊D5—62no—
191#709—64no—
192#709—122no—
193C2×D20C22×D311yescharacteristic
194C2×C4×D5—21no—
195C2×C4×D5—62no—
196C2×Dic10—32no—
197C5×C4⋊C4—21no—
198C5×C4⋊C4—31no—
199C5×C4⋊C4—31no—
200C5×C4⋊C4—62no—
201C5×C4⋊C4—61no—
202C5×C4⋊C4C22×D312yes—
203C5×C22⋊C4—32no—
204C5×C22⋊C4—62no—
205C5×C22⋊C4—62no—
206C42×C5—62no—
207C42×C5—61no—
208C42×C5C22×D311yescharacteristic
209C22⋊Dic5—62no—
210#689—122no—
211#688—62no—
212#685—21no—
213C4⋊Dic5—61no—
214C4⋊Dic5—61no—
215C4⋊Dic5C22×D311yescharacteristic
216#683—62no—
217C2×#168—31no—
218C2×#168—31no—
219C2×#168—122no—
220C2×#168C22×D312yes—
221#556—31no—
222#547—151no—
223#487—151no—
224#485—152no—
225#474—51no—
226#467—152no—
227#462—151no—
228#452—152no—
229C4×SD16—151no—
230C8×D4—151no—
231#416—152no—
232C22×C3×C5C22×D411yescharacteristic
233C2×C5×D3—22no—
234C2×C5×D3—22no—
235C2×C5×D3—22no—
236C2×C3×D5—22no—
237C2×C3×D5—41no—
238C2×C3×D5—41no—
239C3×C4×C5—21no—
240C3×C4×C5—42no—
241C3×C4×C5#13512yes—
242C3×C4×C5C2×#4111yescharacteristic
243C3×C4×C5H(5,𝔽2)11yescharacteristic
244Dic15—42no—
245C3×Dic5—41no—
246C5×Dic3—22no—
247C5×Dic3—41no—
248C5×Dic3—41no—
249C2×C3×Q8—21no—
250C2×C3×D4—51no—
251C22×C3×C4—101no—
252C2×#67—102no—
253C22×Dic3—102no—
254#237—104no—
255#237—202no—
256C2×D12C22⋊D511yescharacteristic
257C2×C4×D3C22×D512yes—
258C2×Dic6—52no—
259C2×Dic6C22×D511yescharacteristic
260C2×C3×C8—52no—
261C3×C4⋊C4—21no—
262C3×C4⋊C4—51no—
263C3×C4⋊C4C22×D512yes—
264C3×C22⋊C4—101no—
265C3×C42C22×D511yescharacteristic
266#214—102no—
267(C2×D3)⋊C4—22no—
268(C2×D3)⋊C4—22no—
269(C2×D3)⋊C4—102no—
270(C2×D3)⋊C4C22⋊D512yes—
271C4⋊Dic3—22no—
272C4⋊Dic3—101no—
273C4⋊Dic3—101no—
274C4⋊Dic3—102no—
275C4⋊Dic3C22⋊D511yescharacteristic
276C4⋊Dic3C22×D511yescharacteristic
277Dic3⋊C4—21no—
278C4×Dic3—21no—
279C4×Dic3—22no—
280C2×C3⋊C8—51no—
281C2×C3⋊C8—51no—
282Dic12—202no—
283D3×C8—202no—
284C23×C5—32no—
285C22×D5—21no—
286C5×Q8—42no—
287C5×Q8—61no—
288C5×Q8—61no—
289C5×D4—62no—
290C2×C4×C5—22no—
291C2×C4×C5—32no—
292C2×C4×C5—62no—
293C2×C4×C5—62no—
294C2×C4×C5—61no—
295C2×C4×C5—61no—
296C2×C4×C5D3×D411yescharacteristic
297C2×C4×C5D3×D411yescharacteristic
298C2×C4×C5C23×D311yescharacteristic
299C22⋊D5—124no—
300C2×Dic5—21no—
301C2×Dic5—62no—
302D20—21no—
303D20#23912yes—
304C4×D5—42no—
305C4×D5—124no—
306Dic10—62no—
307Dic10—62no—
308#168—22no—
309#168—61no—
310#168—61no—
311C2×#41—152no—
312C2×Q16—152no—
313C2×SD16—151no—
314C22×C8—152no—
315C4⋊Q8—151no—
316C4⋊Q8D3×D511yescharacteristic
317#126—62no—
318#124—61no—
319#123—302no—
320C22⋊Q8—32no—
321C22⋊Q8—152no—
322#121—301no—
323C4×Q8—31no—
324C4×Q8—61no—
325C4×D4—31no—
326C4×D4—51no—
327#117—32no—
328C2×C4⋊C4—301no—
329#107—52no—
330#106—151no—
331C4⋊C8—51no—
332#103—152no—
333#103—151no—
334#103—152no—
335#103—152no—
336#102—52no—
337#102—151no—
338C22⋊C8—152no—
339C4×C8—151no—
340C2×C3×C5#54611yescharacteristic
341C2×C3×C5C2×#13511yescharacteristic
342C2×C3×C5#57811yescharacteristic
343C3×D5—42no—
344C5×D3—42no—
345C23×C3—101no—
346C22×D3C2×C22⋊D512yes—
347C3×Q8—42no—
348C3×D4—52no—
349C3×D4—101no—
350C2×C3×C4—22no—
351C2×C3×C4—101no—
352C2×C3×C4—202no—
353C2×C3×C4C2×C22⋊D511yescharacteristic
354C2×C3×C4C2×C22⋊D511yescharacteristic
355C2×C3×C4C23×D511yescharacteristic
356#67—204no—
357C2×Dic3—21no—
358C2×Dic3—21no—
359C2×Dic3—102no—
360C2×Dic3—204no—
361C2×Dic3C2×C22⋊D512yes—
362D12—22no—
363C4×D3—22no—
364C4×D3—22no—
365Dic6—21no—
366Dic6—21no—
367Dic6—102no—
368Dic6—102no—
369C3×C8—102no—
370C3⋊C8—101no—
371C3⋊C8—101no—
372C22×C5—62no—
373C22×C5—62no—
374C22×C5—62no—
375C22×C5C2×D3×D411yescharacteristic
376C2×D5—22no—
377C2×D5—41no—
378C2×D5—41no—
379C4×C5—21no—
380C4×C5—42no—
381C4×C5—62no—
382C4×C5—121no—
383C4×C5—121no—
384C4×C5#91212yes—
385C4×C5C2×#23911yescharacteristic
386C4×C5#100911yescharacteristic
387Dic5—41no—
388Dic5—122no—
389#41—304no—
390#41—602no—
391C2×Q8—21no—
392C2×Q8—31no—
393C2×Q8—152no—
394C2×D4—31no—
395C2×D4—51no—
396C2×D4—302no—
397C22×C4—32no—
398C22×C4—101no—
399C22×C4—302no—
400Q16—602no—
401SD16—302no—
402C2×C8—52no—
403C2×C8—151no—
404C2×C8—151no—
405C2×C8—602no—
406C4⋊C4—21no—
407C4⋊C4—31no—
408C4⋊C4—31no—
409C4⋊C4—51no—
410C4⋊C4—62no—
411C4⋊C4—61no—
412C4⋊C4—302no—
413C4⋊C4—301no—
414C4⋊C4—301no—
415C4⋊C4C2×D3×D512yes—
416C22⋊C4—32no—
417C22⋊C4—62no—
418C22⋊C4—62no—
419C22⋊C4—101no—
420C22⋊C4—302no—
421C42—62no—
422C42—61no—
423C42C2×D3×D511yescharacteristic
424C3×C5#330211yescharacteristic, complemented, Hall
425C22×C3—102no—
426C22×C3—201no—
427C22×C3—201no—
428C22×C3C22×C22⋊D511yescharacteristic
429C2×D3—22no—
430C2×D3—22no—
431C2×D3—22no—
432C3×C4—21no—
433C3×C4—42no—
434C3×C4—201no—
435C3×C4#409612yes—
436C3×C4D5×#4111yescharacteristic
437C3×C4#414911yescharacteristic
438Dic3—22no—
439Dic3—41no—
440Dic3—41no—
441Dic3—202no—
442C2×C5—122no—
443C2×C5#563011yescharacteristic
444C2×C5C2×#91211yescharacteristic
445C2×C5#577911yescharacteristic
446D5—42no—
447C23—32no—
448C23—101no—
449Q8—42no—
450Q8—61no—
451Q8—61no—
452Q8—302no—
453Q8—302no—
454D4—52no—
455D4—62no—
456D4—101no—
457D4—604no—
458C2×C4—22no—
459C2×C4—32no—
460C2×C4—62no—
461C2×C4—62no—
462C2×C4—61no—
463C2×C4—61no—
464C2×C4—101no—
465C2×C4—202no—
466C2×C4—302no—
467C2×C4—604no—
468C2×C4D3×C22⋊D511yescharacteristic
469C2×C4D3×C22⋊D511yescharacteristic
470C2×C4C22×D3×D511yescharacteristic
471C8—102no—
472C8—301no—
473C8—301no—
474C2×C3—202no—
475C2×C3C2×#409611yescharacteristic
476C2×C3#6616311yescharacteristic
477C2×C3#6617211yescharacteristic
478D3—42no—
479C5#8098111yescharacteristic, complemented, Sylow
480C22—62no—
481C22—62no—
482C22—62no—
483C22—102no—
484C22—201no—
485C22—201no—
486C22C2×D3×C22⋊D511yescharacteristic
487C4—21no—
488C4—42no—
489C4—62no—
490C4—121no—
491C4—121no—
492C4—201no—
493C4—602no—
494C4#9149312yes—
495C4D5×#23911yescharacteristic
496C4#9203511yescharacteristic
497C3#1061539511yescharacteristic, complemented, Sylow
498C2—122no—
499C2—202no—
500C2C2×#9149311yescharacteristic
501C2#1175511411yescharacteristic
502C2#1175527711yescharacteristic
subgroupG/H
centerC22C2×D3×C22⋊D5
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC3×C5×C4⋊Q8C22
socleC22×C3×C5C22×D4
prime radical#11752532C2
hypercenterC4⋊Q8D3×D5
hypocenterC3×C5#3302
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C5×#5732C2semidirect
3. #11751194C2semidirect
4. C3×#65376C2semidirect
5. #11747514C2semidirect
6. #11747502C2semidirect
7. #11747447C2semidirect
8. #11747176C2semidirect
9. #11747101C2semidirect
10. C4×#6655C2semidirect
11. #11746998C2semidirect
46. #65376D3semidirect
89. #5732D5semidirect
424. C3×C5#3302semidirect
479. C5#80981semidirect
497. C3#10615395semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C5×#57322
#117525322
#117511942
C3×#653762
#117475142
#117475022
#117474472
#117471762
#117471012
C4×#66552
#117469982
#106153953
#809815